Uncertainty quantification of brake squeal Iistability via surrogate modelling

Noise, vibration and Harshness (NVH) of automotive disc brakes have been an active research topic for several decades. The environmental concerns, on one hand, and the rising customer expectations of their car quality, on the other hand, have made NVH of brakes an important issue for car manufacturers. Of different types of noise and vibration that a brake system may generate, squeal is the main focus of the current study. Brake squeal is an irritating high-frequency noise causing a significant warranty cost to car manufacturers. There are a number of reasons leading to squeal noise either at the end of production or during usage and services. Of these reasons, it is believed that manufacturing variability, several sources of uncertainty (such as friction and contact) and diverse loading cases have the most contribution in this problem. Car manufacturers are then recently encouraged to look into the uncertainty analysis of the brake systems in order to cover the influence of these variations on brake designs. The biggest hurdle in the uncertainty analysis of brakes is the computational time, cost and data storage. In general, stochastic studies are done on the premise of deterministic analyses of a system. As the deterministic analyses of brake squeal instability essentially involve a great deal of computational workload, their stochastic (non-deterministic) analyses will be consequently very expensive. To overcome this issue, the method of surrogate modelling is proposed in this study. Briefly speaking, surrogate modelling replaces an expensive simulation code with a cheap-to-evaluate mathematical predictor. As a result, instead of using the actual finite element model of a brake for statistical analyses, its replacement model will be used alternatively. There are three main advantages in surrogate modelling of brakes. First of all, it paves the way of structural modification of brakes, which are conventionally done for reducing squeal propensity. Secondly, structural uncertainties of a brake design can cost-effectively be propagated onto the results of the stability analysis. Thereafter, instead of making a single design point stable, a scatter of points should meet the stability criteria. Finally, the reliability and robustness of a brake design can be quantified efficiently. These two measures indicate the probability of unstable vibration leading to squeal noise for a brake design. Accordingly, car manufacturers will be able to estimate the cost of warranty claims which may be filed due to this particular issue. If the probability of failure which is calculated for squeal propensity is significant, surrogate modelling helps come up with a solution during the design stage, before cars go into production. In brief, two major steps must be taken toward constructing a surrogate model: making a uniform sampling plan and fitting a mathematical predictor to the observed data. Of different sampling techniques, Latin hypercube sampling (LHS) is used in this study in order to reduce the amount of computational workload. It is worth mentioning that the original LHS does not enforce the uniformity condition when making samples. However, some modifications can be applied to LHS in order to improve the uniformity of samples. Note that the uniformity of samples plays a crucial role in the accuracy of a surrogate model. A surrogate model, in fact, is built on the premise of the observations which are made over a design space. Depending on the nonlinearity of the outputs versus the input variables and also depending on the dimensions of a design space, different mathematical functions may be used for a surrogate predictor. The results of this study show that Kriging function brings about a very accurate surrogate model for the brake squeal instability. In order to validate the accuracy of surrogate models, a number of methods are reviewed and implemented in the current study. Finally, the validated surrogate models are used in place of the actual FE model for uncertainty quantification of squeal instability. Apart from surrogate modeling, a stochastic study is conducted on friction-induced vibration. Statistics of complex eigenvalues of a simplified brake models are studied under the influence of variability and uncertainty. For this purpose, the 2nd order perturbation method is extended to be applicable on an asymmetric system with non-proportional damping. The main advantage of this approach is that the statistics of complex eigenvalues can be calculated in just one run, which is massively more efficient than the conventional techniques of uncertainty propagation that use a large number of simulations to determine the results

Computational nonlinear vibration analysis for distributed geometrical nonlinearities in structural dynamics

The demand to reduce the impact of aviation on the environment is leading jet engine manu- facturers to increase the fuel and propulsion efficiency of the engines. This in turn is pushing materials to their physical limits by undergoing increasingly higher thermo-mechanical loads. In this regime, blades and other engine components are subjected to large deforma- tions generating nonlinearities that activate new failure mechanisms not dealt with before. Therefore, vibration analysis is essential to develop new methodologies for the accurate prediction of components’ behaviour. This research focuses on investigating the effect of the distributed geometric nonlinearities and rotational speed on the dynamic behaviour of three-dimensional structures. The Green-Lagrange strain measures are employed in this research to express the nonlinear relationship between the displacement and the strain. The nonlinear algorithms used for the numerical simulations are developed based on the Finite Element Method combined with the Harmonic Balance method. The complex geometries are discretised by using the geometric exact three-dimensional solid elements. The forced response functions and the backbone curves for the steady-state response of the nonlinear system can be computed. The research aims to develop and validate methodologies for the identification and control of undesired vibration modes which will inform new design choices. Finite element modelling of the blades generally involves an immense number of degree-of-freedoms, which could be infeasible to compute. The reduced order modelling (ROM) techniques are crucial for achieving an accurate prediction of the nonlinear behaviour in an efficient way. Detailed computation strategies for the intrusive ROM methods are delivered. ROM techniques based on the linear and nonlinear mapping between the full model and the reduced basis are presented. The capabilities and limitations of both methods are assessed. The projection method based on the linear eigenmodes only has a slow converge to the full system. On the other hand, the quadratic manifold method with the static modal derivatives involved in the reduced coordinates provides a fast convergence.Open Acces

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

MultiBody Dynamic Analysis of a 3D Synchronizer Model

L'abstract è presente nell'allegato / the abstract is in the attachmen

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal